New Techniques for Computing Geometric Index

TL;DR

This paper presents new, unified techniques for computing the geometric index of links in solid tori, simplifying previous methods and enabling calculations for previously unknown cases.

Contribution

The paper introduces general techniques for computing the geometric index in solid tori, unifying and simplifying prior ad hoc methods and extending computability to new examples.

Findings

New techniques simplify geometric index calculations.

Method applies to complex links previously difficult to analyze.

Facilitates easy computation when solid torus is divided into chambers.

Abstract

We introduce \textcolor{red}{general} new techniques for computing the geometric index of a link $L$ in the interior of a solid torus $T$. These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples \textcolor{red}{ and allow the simple computation of geometric index for new examples where the index was not previously known}. The geometric index measures the minimum number of times any meridional disc of $T$ must intersect $L$. It is related to the algebraic index in the sense that adding up signed intersections of an interior simple closed curve $C$ in $T$ with a meridional disc gives $\pm$ the algebraic index of $C$ in $T$. One key idea is introducing the notion of geometric index for solid chambers of the form $B^2\times I$ in $T$. After that we prove that if a solid torus can be divided into solid chambers by meridional discs in a specific \textcolor{red}{(and often easy to obtain)} way, then the geometric index can be easily computed.